Suppose you want to design an auction for the following type of situation: you have two identical…
Suppose you want to design an auction for the following type of situation: you have two identical copies of a valuable object, and there are four potential buyers for the object. Each potential buyer j wants at most one copy and has a value vj for either copy
You decide to design the auction by analogy with the way in which we derived the single-item ascending-bid (English) auction from the general procedure for matching markets. In the present case, as there, you want to create a bipartite graph that encodes the situation and then see what prices the bipartite-graph auction procedure comes up with.
(a) Describe how this construction would work using an example with four potential buyers. In creating your example, first choose specific valuations for the potential buyers, and then show how the auction proceeds and what the market-clearing prices are.
(b) In the case of the single-item auction, the bipartite-graph procedure yielded the simple rule from the ascending-bid (English) auction: sell to the highest bidder at the second-highest price. Describe in comparably simple terms what the rule is for the current case of two identical items (i.e., your description should not involve the terms “bipartite,” “graph,” or “matching”).